High order modulation protograph codes

ABSTRACT

Digital communication coding methods for designing protograph-based bit-interleaved code modulation that is general and applies to any modulation. The general coding framework can support not only multiple rates but also adaptive modulation. The method is a two stage lifting approach. In the first stage, an original protograph is lifted to a slightly larger intermediate protograph. The intermediate protograph is then lifted via a circulant matrix to the expected codeword length to form a protograph-based low-density parity-check code.

STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH OR DEVELOPMENT

The invention described herein was made in the performance of work undera NASA contract, and is subject to the provisions of Public Law 96-517(35 USC 202) in which the Contractor has elected to retain title.

BACKGROUND

1. Field

The present disclosure relates to constructing low-density parity-check(LDPC) codes from small template graphs called protographs. More inparticular, it relates to methods for designing protographs thataccommodate modulation levels above simple binary coding.

2. Description of Related Art

DEFINITIONS

As known to the person skilled in the art and as also mentioned in U.S.Pat. No. 7,343,539 incorporated herein by reference in its entirety, alow-density parity-check (LDPC) code is a linear code determined by asparse parity-check matrix H having a small number of 1 s per column.The code's parity-check matrix H can be represented by a bipartiteTanner graph wherein each column of H is represented by a transmittedvariable node, each row by a check node, and each “1” in H by a graphedge connecting the variable node and check node that correspond to thecolumn-row location of the “1”. The code's Tanner graph may additionallyhave non-transmitted variable nodes. Each check or constraint nodedefines a parity check operation. Moreover, the fraction of atransmission that bears information is called the rate of the code. AnLDPC code can be encoded by deriving an appropriate generator matrix Gfrom its parity-check matrix H. An LDPC code can be decoded efficientlyusing a well-known iterative algorithm that passes messages along edgesof the code's Tanner graph from variable nodes to check nodes andvice-versa until convergence is obtained, or a certain number ofiterations is reached.

A protograph is a Tanner graph with a relatively small number of nodes,connected by a small number of edges, allowing multiple parallel edgesbetween two nodes. Each edge is a type. Thus, the number of total edgesrepresents the number of types that exists in the protograph. Aprotograph code (an equivalent LDPC code) is a larger derived graphconstructed by applying a “copy-and-permutation” operation on aprotograph. The protograph is copied N times, then a large LDPC codegraph is obtained by permuting N variable-to-check pairs (edges),corresponding to the same edge type of the original protograph. Theresulting protograph code has N times as many nodes as the protograph,but it has the same rate and the same distribution of variable and checknode degrees. Thus, the degree distribution of a protograph-based LDPCcode is the same as that of the protograph. A simple example of aprotograph is shown in FIG. 1. This graph consists of 3 variable nodesand 2 check nodes that are interconnected by 7 different edge(variable-to-check) types. The derived graph is constructed byreplicating the protograph 3 times, and permuting variable-to-checkpairings within the same edge type of the protograph. A protograph canalso be represented by a proto-matrix. For example, the proto-matrix ofthe protograph shown in the left portion of FIG. 1 has the form shown inEq. 1 below:

$\begin{matrix}{H_{proto} = \begin{pmatrix}1 & 1 & 1 \\2 & 1 & 1\end{pmatrix}} & {{Eq}.\mspace{14mu} 1}\end{matrix}$where the rows and columns represent the check nodes and variable nodesin the graph respectively. The elements in the matrix are the number ofparallel edges that connect the variable node and the check nodeassociated with their positions.

One way to construct larger codes is to replace each integer j in theproto-matrix with sum of j different N×N permutation matrices. If theinteger is 1, it would be replaced with 1 N×N permutation matrix.Another way is to first expand the protograph by a small factor suchthat the H matrix for the resulting expanded protograph does not containany integer larger than 1. Then each integer 1 in the H matrix will bereplaced with an N×N permutation matrix. The assigned N×N permutationscan be any type of permutations, including circulant permutations. AnN×N identity matrix can be regarded as a trivial permutation matrix. Ifthe N×N permutation is a circulant permutation, then define matrix X asone circular shift of the identity matrix to the left. Using thisnotation, then all circulant permutations can be represented by X^(i),for i=0, 1, . . . , (N−1). The selection and assignment of permutationmatrices to construct N times larger LDPC code can be based on enlargingthe smallest loop size in the expanded graph. Other optimization methodsinclude Progressive Edge Growth (PEG) for circulant matrices (see, forexample X. Hu, E. Eleftheriou, and D. M. Arnold, “Regular and irregularprogressive edge-growth tanner graphs,” IEEE Transactions on InformationTheory, Vol. 51, issue 1, 2005, pp. 386-398, hereinafter X. Hu, et al.),or the Approximate Cycle Extrinsic message degree (ACE) algorithm (see,for example, T. Tian, C. R. Jones, J. D. Villasenor, and R. D. Wesel,“Selective avoidance of cycles in irregular LDPC code construction,”IEEE Transactions on Communications, Volume 52, Issue 8, 2004, pp.1242-1247, or T. Tian, C. R. Jones, J. D. Villasenor, and R. D. Wesel,“Construction of irregular LDPC codes with low error floors,” IEEEInternational Conference on Communications, 2003, Vol. 5, pp.3125-3129). So, in the case of circulant permutations, theseoptimization algorithms can be applied to obtain powers of X in the Hmatrix of the expanded protograph.

The rate of a protograph is defined to be the lowest (and typical) rateof any LDPC code constructed from that protograph. All LDPC codesconstructed from a given protograph have the same rate except forpossible check constraint degeneracies, which can increase (but neverdecrease) this rate and typically occur only for very small codes. Sincethe protograph serves as a blueprint for the Tanner graph representingany LDPC code expandable from that protograph, it also serves as ablueprint for the routing of messages in the iterative algorithm used todecode such expanded codes through a permutation per each edge type. Therate of a protograph code is computed as shown below in Eq. 2:

$\begin{matrix}{r = \frac{n - m}{n - n_{0}}} & {{Eq}.\mspace{14mu} 2}\end{matrix}$where n and m are number of variable nodes and check nodes respectivelyin the protograph, n₀ is number of punctured (untransmitted) variablenodes.

Excluding check nodes connected to degree-1 variable nodes, applicantshave proved that the number of degree-2 nodes should be at most one lessthan the number of check nodes provided that no loop exists in the graphbetween degree-2 nodes and the checks connected to these nodes for aprotograph to have the linear minimum distance property. A givenprotograph is said to have the linear minimum distance property if thetypical minimum distance of a random ensemble of arbitrarily large LDPCcodes built from that protograph grows linearly with the size of thecode, with linearity coefficient which can be denoted by δ_(min)>0 (seeS. Abu-Surra, D. Divsalar and W. Ryan, “On the existence of typicalminimum distance for protograph-based LDPC codes,” in Information Theoryand Applications Workshop (ITA), January 2010, pp. 1-7).

The iterative decoding threshold of a given protograph is similarlydefined with respect to this random ensemble of LDPC codes as the lowestvalue of signal-to-noise ratio for which an LDPC decoder's iterativedecoding algorithm will find the correct codeword with probabilityapproaching one as the size of an LDPC code built from that protographis made arbitrarily large. Iterative decoding thresholds can becalculated by using a reciprocal channel approximation (D. Divsalar, S.Dolinar, C. R. Jones, and K. Andrews, “Capacity approaching protographcodes,” IEEE J. Select. Areas Communication, vol. 27, no. 6, pp.876-888, August 2009, hereinafter D. Divsalar, et al.) or the PEXITmethod (see, for example, G. Liva and M. Chiani, “Protograph LDPC codesdesign based on EXIT analysis,” in Proc. IEEE GLOBECOM, November 2007,pp. 3250-3254, hereinafter G. Liva, et al.). Thresholds can be loweredeither by using precoding (a subgraph of the protograph with degree-1nodes, check nodes connected to these degree-1 nodes, and all edgesconnected to these checks and other variable nodes with at least onepunctured node) or through the use of at least one very high-degree nodein the base protograph. A protograph is said to have a low iterativedecoding threshold if its threshold is close to the capacity limit forits rate.

A family of protographs of different rates is said to be rate-compatiblewith embedding or embedded rate-compatible if the protographs fordifferent rates produce embedded codewords with the same informationblock-length. In other words, with the same input, the codeword of ahigh-rate code is embedded into the codeword of a lower-rate code. Thisproperty makes this family of protographs suitable for Hybrid AutomaticRepeat Request (HARQ) applications.

Coded modulation is a method of communication where the messages areencoded with an error correcting code, and the resulting coded bits aremapped to modulation symbols such as BPSK, M-PSK, or M-QAM modulations.The overall efficiency of the coded modulation is measured by spectralefficiency, whose unit is bits/seconds/Hz.

Bit-interleaved coded modulation (BICM) is a technique that allowsrelatively simple design of bandwidth-efficient coded modulationsystems. A general BICM system is shown in FIG. 2. The informationsequence is encoded by an LDPC encoder to get a coded bit sequence.Depending on the fading regime, the coded sequence may bebit-interleaved before being sent to a modulator. The M-ary modulatormaps m=log₂M coded bits at a time to a complex symbol chosen from anM-ary constellation χ. The discrete-time baseband channel model can bewritten as shown in Eq. 3 below:y _(t) =hx _(t) +w _(t)   Eq. 3where t is the discrete time index, y_(t) is the received signal, x_(t)is the transmitted symbol, h=h_(I)+ih_(Q) is the zero mean complexGaussian distributed fading coefficient with the variance of ½ in eachdimension and w_(t) is a complex white Gaussian noise sample with thezero mean and the variance per dimension of σ²=N₀/2. It is assumed thatthe channel state information is available at the receiver, i.e. thereceiver can estimate the channel coefficient h.

The MAP symbol-to-bit metric calculator will compute the symbol-to-bitmetrics based on the received symbol y from the channel. These bitmetrics are passed to a decoder, which employs the iterative beliefpropagation algorithm. When an interleaver is used in the transmitter,the bit metrics in the receiver are de-interleaved before being passedto the LDPC decoder. Let χ_(b) ^(i) be the subset of all the signalpoints x∈ χ hole label has value b∈ {0, 1} in position i. Thesymbol-to-bit metric is computed by the MAP calculator at each time tofeed into the iterative decoder is given as shown below in Eq. 4:

$\begin{matrix}\begin{matrix}{L_{i} = {L( {b_{i}❘y} )}} \\{= {\ln\frac{\sum\limits_{x \in \chi_{0}^{i}}\;{\exp( {{- \frac{1}{2\;\sigma^{2}}}{{y - {hx}}}^{2}} )}}{\sum\limits_{x \in \chi_{1}^{i}}\;{\exp( {{- \frac{1}{2\;\sigma^{2}}}{{y - {hx}}}^{2}} )}}}}\end{matrix} & {{Eq}.\mspace{14mu} 4}\end{matrix}$for i=1, . . . , m, where M=2^(m). The BICM capacity with perfect CSIand uniform inputs is given by Eq. 5 as shown below:

$\begin{matrix}{C = {m - {\sum\limits_{i = 1}^{m}\;{E_{b,y,h}\lbrack {\log_{2}\frac{\sum\limits_{x \in \chi}\;{p( {{y❘x},h} )}}{\sum\limits_{x \in \chi_{b}^{i}}\;{p( {{y❘x},h} )}}} \rbrack}}}} & {{Eq}.\mspace{14mu} 5}\end{matrix}$

BICM capacity is the sum of mutual information of m parallel channels asshown in Eq. 6 below:

$\begin{matrix}{C = {\sum\limits_{i = 1}^{m}\;{I( {L_{i},b_{i}} )}}} & {{Eq}.\mspace{14mu} 6}\end{matrix}$This mutual information I(L_(i), b_(i)) can be computed by using a MonteCarlo simulation as shown in Eq. 7 and Eq. 8 below:I(L _(i) , b _(i))=1−E[log ₂(1+exp(−(1−2b _(i))L _(i)))]  Eq. 7

$\begin{matrix}{{I( {L_{i},b_{i}} )} = {1 - {\frac{1}{N}{\sum\limits_{n = 1}^{N}\;{\log_{2}( {1 + {\exp( {{- ( {1 - {2\; b_{i,n}}} )}L_{i,n}} )}} )}}}}} & {{Eq}.\mspace{14mu} 8}\end{matrix}$

where N is the number of coded modulation symbols, b_(i) and L_(i) arethe bit and log-likelihood random variables respectively for the level iof coded modulation, and b_(i, n) and L_(i, n) are respectively the bitand log-likelihood values at level i for the n-th symbol.

The BICM approach has proven to be powerful and capacity-approaching forfading channels with applications of low-density parity-check (LDPC)codes. However, many of the previous works design a particular code witha specific modulation scheme. It is desirable to find a general schemethat can support multiple rates and multiple modulation schemes withinthe context of BICM.

SUMMARY

Described herein are embodiments that provide for digital communicationcoding methods for designing protograph-based BICM that is general andapplies to any modulation. The iterative decoding thresholds of theprotograph codes while mapped to higher order modulations arecalculated. This general coding framework can support not only multiplerates but also adaptive modulation. Certain families of protograph codesare shown to achieve a threshold within a gap of approximately 0.2-0.3dB of BICM capacity limit across a wide range of rates and modulations.

BRIEF DESCRIPTION OF THE SEVERAL VIEWS OF THE DRAWINGS

FIG. 1 shows an example of a protograph and demonstrates acopy-and-permute operation on the protograph to obtain a larger graph.The protograph also can be represented by a protomatrix whose elementspresent the number of edges connecting a variable node (column) to acheck node (row) in the protograph.

FIG. 2 shows a block diagram of a bit-interleaved coded modulationsystem.

FIG. 3 shows the AR4JA family of protographs with rates ½ and higher.

FIG. 4 shows a rate ½ protograph which proves a better iterativedecoding threshold.

FIG. 5 shows the protomatrix corresponding to the protograph shown inFIG. 4.

FIG. 6 shows the protomatrices of high-rate (>½) protographs that areextended from the rate-½ protomatrix shown in FIG. 5.

FIG. 7 shows the protomatrices of rate-compatible family built fromextending the rate-¾ protograph whose protomatrix is shown in FIG. 6.

FIG. 8 shows a rate-½ AR4JA protograph.

FIG. 9 shows a rate-½ AR4JA protograph after lifting by a factor of 4with a mapping for 16 QAM.

FIG. 10 shows the iterative decoding thresholds from methods describedherein with iterative coding thresholds obtained from other mappingmethods with rate-½ AR4JA protograph and 16QAM in AWGN channel.

FIG. 11 shows the iterative coding thresholds of AR4JA codes shown inFIG. 3 utilizing the described BICM methods in Rayleigh faded channels.

FIG. 12 shows the iterative coding thresholds of another family of codeswhose protomatrices are shown in FIG. 5 and FIG. 6 utilizing thedescribed BICM methods in Rayleigh faded channels.

FIG. 13 shows the iterative coding thresholds of rate-compatible familyof codes whose protomatrices are shown in FIG. 7 utilizing the describedBICM methods in Rayleigh faded channels.

FIG. 14 shows 8PSK with Gray labeling.

FIG. 15 shows 16QAM with Gray labeling.

FIG. 16 shows the performance of rate-½ AR4JA protograph and the rate-½protograph whose protomatrix is shown in FIG. 7 with 16QAM in Rayleighchannel and 16 k information bits.

DETAILED DESCRIPTION

The exemplary embodiments described below show the application of threefamilies of protograph codes with embodiments of the present invention.Note, however, that the present invention is not limited to theprotograph codes presented below. Descriptions of these protograph codesare presented to aid in understanding of the invention. Those skilled inthe art will understand that other protograph codes are within the scopeof the invention. The families of protograph codes presented belowinclude: AR4JA codes, discussed in more detailed in D. Divsalar, et al.,and two other families of protograph codes presented in T. V. Nguyen, A.Nosratinia, and D. Divsalar, “The design of rate compatible protographLDPC codes,” IEEE Trans. Commun., 2011, hereinafter T. V. Nguyen, et al.The AR4JA codes have been recommended for the space communication forthe code rate-½ or higher. Based on the same AR4JA protograph structure,T. V. Nguyen, et al. provide a design procedure that gave a family ofnew codes with an improvement of about 0.25 dB in terms of iterativedecoding thresholds in AWGN channels compared with that of AR4JAprotograph codes. The third family that consists of rate-compatiblecodes is built from extending a rate-¾ code in the second family. Thisthird family of codes gives rise to coded modulations that can supporthybrid ARQ applications, among others.

The family of AR4JA protograph codes is plotted in FIG. 3. In FIG. 3,the dark circles represent transmitted variable nodes, the white circleis a punctured node and the circles with a plus sign are parity checknodes. The graph contains 4+2n transmitted variable nodes and 3 checknodes that is equivalent to code rate

${R = \frac{n + 1}{n + 2}},$with n=0, 1, . . . .

A protograph can also be represented by a protomatrix. For example, therate-½ AR4JA protograph (n=0) has a protomatrix in the form shown in Eq.9 below:

$\begin{matrix}{H_{1/2} = \begin{pmatrix}1 & 2 & 0 & 0 & 0 \\0 & 3 & 1 & 1 & 1 \\0 & 1 & 2 & 2 & 1\end{pmatrix}} & {{Eq}.\mspace{14mu} 9}\end{matrix}$where the rows and columns represent the check nodes and variable nodesin the graph respectively. The elements in the matrix are the number ofparallel edges that connect the variable node and the check nodeassociated with their positions. The family of AR4JA codes with rate

$R = \frac{n + 1}{n + 2}$has a protomatrix in the form shown in Eq. 10 below:

$\begin{matrix}{H_{\frac{n + 1}{n + 2}} = ( {H_{\frac{n}{n + 1}}❘\begin{matrix}0 & 0 \\3 & 1 \\1 & 3\end{matrix}} )} & {{Eq}.\mspace{14mu} 10}\end{matrix}$where n=1, 2, . . . The embedded codes with the rate of

$R = \frac{n + 1}{n + 2}$are built by extending the rate −½ code with the protomatrix of Eq.1.The structure of the rate-½ AR4JA code has 5 variable nodes (onepunctured node) and 3 check nodes. For the same rate-½ structure with aslightly larger graph which contains 7 variable nodes (one puncturednode) and 4 check nodes, the protograph of a better code in terms ofiterative decoding threshold is shown in FIG. 4. Its correspondingprotomatrix is shown in FIG. 5. The threshold of this code is 0.395 dBas shown in T. V. Nguyen, et al, which shows a gap of 0.208 dB ofcapacity, less than that of the rate ½ AR4JA code of Eq. 9 as reportedin D. Divsalar, et al. Other high rate protographs which are also builtby extending this rate-½ protograph have protomatrices shown in FIG. 6.

The third family of rate-compatible protograph codes has proto-matricesthat are shown in FIG. 7. This family is built from adding equal numberof variable (column) and check (row) nodes to the rate-¾ protographwhose protomatrix is shown in FIG. 6. The protograph codes in thisfamily have the same information block-lengths, thus are suitable forHybrid ARQ applications.

An iterative decoding threshold of a protograph is the minimum channelquality that supports reliable iterative decoding of asymptoticallylarge LDPC codes built from the protograph. T. J. Richardson, M. A.Shokrollahi, and R. L. Urbanke, “Design of capacity-approachingirregular low-density parity-check codes,” IEEE Trans. Inform. Theory,vol. 47, no. 2, pp. 619-637, February 2001, describe computation ofthresholds of LDPC code ensembles by using density evolution. S. tenBrink, “Convergence behavior of iteratively decoded parallelconcatenated codes,” IEEE Trans. Commun., vol. 49, no. 10, pp.1727-1737, October 2001, describe computation of thresholds of LDPC codeensembles by extrinsic information transfer (EXIT) chart techniques.However, the general EXIT chart cannot be applied to compute thethresholds of protograph codes due to its inability of computingthresholds of graphs that have degree-1 variable, punctured variablenodes or protographs with same degree distributions and differentthresholds. See, for example, G. Liva, et al. To solve these problems,G. Liva, et al. proposed the PEXIT method that is simple and allows forthe computation of thresholds of protographs even more accurately thanthe method of reciprocal channel approximation (RCA) described in D.Divsalar, et al., cited above. Thresholds of the AR4JA family ofprotograph codes plotted in FIG. 3 and the other family of protographswere discussed in T. V. Nguyen, et al., cited above.

As discussed above, protograph codes can be represented by a small graphwith only a few variable nodes and check nodes. For the purposes ofBICM, the binary variable nodes of the code must be mapped to thebit-levels of the modulation. The iterative decoding threshold of theoverall coded modulation scheme naturally depends in part on theprotograph code, but also on the mapping between the code and themodulation levels.

In D. Divsalar and C. Jones, “Protograph based low error floor LDPCcoded modulation,” in Proc. IEEE MILCOM, October 2005, pp. 378-385 Vol.1, hereinafter Divsalar & Jones, Divsalar and Jones proposed a mappingalgorithm based on Variable Degree Matched Mapping (VDMM). The idea isto directly assign protograph variable nodes in proportion to mutualinformation of coded bits in the modulation symbol. This is a well-knownwater-filling problem. However, in Divsalar & Jones, variable nodes in aprotograph are directly assigned to the bits of a modulation symbol.This limits the coded modulation that can be designed with this andsimilar method to only one particular coded modulation for eachprotograph. For example, FIG. 8 shows the rate-½ AR4JA protograph withtransmitted variable nodes indexed by V_(i), i=1, . . . , 4. For aspecific 16 QAM modulation, the method of Divsalar & Jones is proposedwith the mapping {b0, b1, b2, b3}={V2, V4, V1, V3}.

An exemplary embodiment of the present invention provides a generalmethod that can work for any protograph structure and modulation leveland can achieve BICM-liked performance. More specifically, thisexemplary embodiment provides a method to map any protograph to anymodulation level, unlike the previous art. Exemplary embodiments use atwo-stage lifting approach. Assuming that the desired modulation isM-ary and the original protograph has n transmitted variable nodes, thefirst step is to start with a smaller lifting of the original protographby a factor of m=log₂M to a slightly larger protograph. This protographtransformation may be accomplished using the PEG algorithm disclosed inX. Hu, et al., or other such techniques. An example is shown in FIG. 9,where an intermediate protograph is constructed from m=4 original AR4JAprotographs. This intermediate protograph is then mapped to n=4modulation symbols as follows: each labeling bit position from all the nmodulation symbols is mapped to the variable nodes of one of the planesin the intermediate protograph. For example, four b₀ bits from fourmodulation symbols are mapped to the top plane, four b₁ bits are mappedto the next plane, etc.

Now, the mapping between coding and modulation is complete within arelatively small “intermediate” protograph. The advantage of thisintermediate protograph is that it is small enough to allowoptimizations, but it also has enough degrees of freedom to provide agood mapping. Now this intermediate protograph is lifted (via acirculant matrix) to the expected codeword length to form aprotograph-based LDPC code.

Embodiments of the present invention may be contrasted with VDMM interms of flexibility. In VDMM as it is proposed, the coded bits in eachprotograph correspond to the symbol bits in one transmitted symbol.Thus, for example, a 4-variable node protograph naturally corresponds to16-QAM. With the proposed framework, however, it is very easy to use anyprotograph together with any modulation. All that is needed is toproduce the right intermediate protograph.

FIG. 10 compares the iterative decoding thresholds of embodimentsaccording to the present invention compared with Divsalar & Jones. Bynecessity this table is small, as Divsalar & Jones as mentioned above,naturally correspond to only one type of modulation. The iterativedecoding threshold of embodiments according to the present invention isslightly better than that of Divsalar & Jones. Note, however, thatembodiments according to the present invention are not simply directedat producing the absolutely smallest threshold for the specific case of16QAM. Embodiments according to the present invention expand the horizonof available designs, as previous designs only allowed the use ofmodulations whose bit-levels were the same as the number of nodes in aprotograph, for example, a protograph with four transmitted nodes couldpreviously be mapped only to a 16-QAM or 16-PSK modulation. As discussedbelow, the threshold gap to capacity provided by embodiments accordingto the present invention is very good over a large spectrum of rates andmodulations.

Presented below are the results obtained by using the PEXIT algorithmbriefly discussed above to compute iterative decoding thresholds.Computation of iterative decoding thresholds for protograph codedmodulation using the PEXIT algorithm is similar to that discussed in G.Liva, et al., for AWGN channel, except the initialization step. Insteadof using the exact equation, as in the initialization step discussed inG. Liva, et al., the mutual information for coded bits is computed usingthe Monte Carlo method shown in Eqs. 7 and 8 above.

The iterative decoding thresholds of protograph-based LDPC coded BICMembodiments according to the present invention are shown tablespresented in FIG. 11, FIG. 12 and FIG. 13. From FIG. 11, AR4JAprotograph-based coded BICM can operate within about 0.7 dB to the BICMcapacity limits. On the other hand, from FIG. 12, the family ofprotograph codes reported in T. V. Nguyen, et al. can operate withinabout 0.2-0.3 dB to the BICM capacity limit, which shows an improvementof about 0.4 dB compared with that of the AR4JA family. From FIG. 13,the family of rate-compatible protographs reported in T. V. Nguyen, etal., can operate within about 0.1-0.2 dB to the BICM capacity limit.These observations match with the results reported in T. V. Nguyen, etal. which provided the coding thresholds in AWGN channel. Discussedbelow is the generation of protograph codes for three modulation schemeswith Gray labeling, i.e. Quadrature Phase Shift Keying (QPSK), 8-aryPhase Shift Keying (8PSK) and 16-ary Quadrature Amplitude Modulation(16QAM) and the calculated the iterative decoding thresholds for thoseprotographs. FIG. 14 shows 8PSK with Gray labeling and FIG. 15 shows16QAM with Gray labeling.

The protograph codes are built from protographs discussed above in twolifting steps. First, the protograph is lifted by a small factor inorder to accommodate all modulation schemes of the group of modulationschemes for which protograph codes are to be generated, e.g., a groupconsisting of QPSK, 8PSK, and 16QAM. In the case of three abovemodulation schemes, the protograph is lifted by a factor of 12 which isa common divisor of 2, 3 and 4 bits . This first lifting procedure maybe implemented by using the progressive edge growth (PEG) algorithm(such as discussed in X. Hu, et al.) in order to remove all multipleparallel edges. Other protograph lifting procedures known in the art mayalso be used. Secondly, the intermediate protograph is furthercirculantly lifted to an expected codeword length depending on differentapplications.

In order to support multiple rates, one can begin with the highest-rate(⅘) protograph. Since the protographs discussed above and shown in FIG.3, FIG. 4, FIG. 5 and FIG. 6 are embedded, the parity-check matrix oflower rate can be obtained by removing certain columns from codes ofhigher rate. To decode the lower-rate codewords, the missing coded bitsare replaced by erasure at the decoder. Thus the family of protographcodes can be implemented within a common encoder/decoder structure andat the same time being able to support adaptive modulation schemes aswell. In order to support rate-compatible codes whose protomatrices areshown in FIG. 7, one can begin with the lowest rate (0.45) protograph.Other higher-rate codes are decoded by replacing missing parity bits byerasures at the decoder, therefore the same decoder can be used for therate-compatible codes.

The performance of the rate-½ protograph code reported in T. V. Nguyen,et al., whose protomatrix is shown in FIG. 7 as well as that of rate-½AR4JA code transmitted with 16QAM in Rayleigh channel is plotted in FIG.16 with the information block-length of 16 k. Both these codedmodulations perform within 1 to 1.2 dB of their capacity limit at 10⁻⁶FER.

The foregoing Detailed Description of exemplary and preferredembodiments is presented for purposes of illustration and disclosure inaccordance with the requirements of the law. It is not intended to beexhaustive nor to limit the invention to the precise form or formsdescribed, but only to enable others skilled in the art to understandhow the invention may be suited for a particular use or implementation.The possibility of modifications and variations will be apparent topractitioners skilled in the art.

No limitation is intended by the description of exemplary embodimentswhich may have included tolerances, feature dimensions, specificoperating conditions, engineering specifications, or the like, and whichmay vary between implementations or with changes to the state of theart, and no limitation should be implied therefrom. In particular it isto be understood that the disclosures are not limited to particularcompositions or biological systems, which can, of course, vary. Thisdisclosure has been made with respect to the current state of the art,but also contemplates advancements and that adaptations in the futuremay take into consideration of those advancements, namely in accordancewith the then current state of the art. It is intended that the scope ofthe invention be defined by the Claims as written and equivalents asapplicable. It is also to be understood that the terminology used hereinis for the purpose of describing particular embodiments only, and is notintended to be limiting. Reference to a claim element in the singular isnot intended to mean “one and only one” unless explicitly so stated. Asused in this specification and the appended claims, the singular forms“a,” “an,” and “the” include plural referents unless the content clearlydictates otherwise. The term “several” includes two or more referentsunless the content clearly dictates otherwise. Unless defined otherwise,all technical and scientific terms used herein have the same meaning ascommonly understood by one of ordinary skill in the art to which thedisclosure pertains.

Moreover, no element, component, nor method or process step in thisdisclosure is intended to be dedicated to the public regardless ofwhether the element, component, or step is explicitly recited in theClaims. No claim element herein is to be construed under the provisionsof 35 U.S.C. Sec. 112, sixth paragraph, unless the element is expresslyrecited using the phrase “means for . . . ” and no method or processstep herein is to be construed under those provisions unless the step,or steps, are expressly recited using the phrase “comprising step(s) for. . . ”

A number of embodiments of the disclosure have been described.Nevertheless, it will be understood that various modifications may bemade without departing from the spirit and scope of the presentdisclosure. Accordingly, other embodiments are within the scope of thefollowing claims.

What is claimed is:
 1. A digital communication coding method comprising:selecting a first protograph having a first protograph structure;lifting the first protograph to a second protograph by a selectedfactor; and lifting the second protograph to a third protograph via acirculant matrix, wherein the third protograph provides a protographcode having a desired codeword length, and wherein the selected factoris m, and m is calculated by the equation as follows:m=log₂M where M is equal to the number of modulation symbols in asignal.
 2. The digital communication coding method according to claim 1,wherein the signal is modulated as quadrate phase-shifted, 8-ary phaseshift keying, or 16-ary quadrature amplitude modulated.
 3. The digitalcommunication coding method according to claim 1, wherein the firstprotograph has n variable nodes and the second protograph has multipleplanes and each labeling bit position from each modulation symbol ismapped to a variable node of one of the planes of the second protograph.4. A digital communication coding method comprising: selecting a firstprotograph having a first protograph structure; lifting the firstprotograph to a second protograph by a selected factor; and lifting thesecond protograph to a third protograph via a circulant matrix, whereinthe third protograph provides a protograph code having a desiredcodeword length, and wherein the coding method supports signals havingmultiple numbers of modulation symbols in a signal, wherein the multiplenumbers are represented by M₁, M₂, . . . , M_(n) and the selected factorcomprises a common divisor from the solution of the equations shownbelow for m₁, m₂ . . . m_(n):m₁=log₂M₁, m₂=log₂M₂, . . . , m_(n)=log₂M_(n).
 5. The digitalcommunication coding method according to claim 1, wherein the firstprotograph is lifted to second protograph by a progressive edge-growthalgorithm.
 6. The digital communication coding method according to claim1, wherein the first protograph comprises a protograph from AR4JA codes.7. The digital communication coding method according to claim 1, whereinthe first protograph comprises a rate-compatible protograph.
 8. Thedigital communication coding method according to claim 1, wherein thefirst protograph comprises a protograph-based LDPC convolutional code.9. The digital communication coding method according to claim 1, whereinthe selected factor is selected to accommodate all modulation schemeswithin a group of modulation schemes.
 10. A digital communication systemcomprising: a low density parity check encoder; a bit interleaver; amapper and modulator; wherein the low density parity check encoder codesbinary data with a code having a selected codeword length, and the codeis generated by: selecting a first protograph having a first protographstructure; lifting the first protograph to a second protograph by aselected factor; and lifting the second protograph to a third protographvia a circulant matrix, wherein the third protograph provides aprotograph code having the selected codeword length, and wherein theselected factor is m, and m is calculated by the equation as follows:m=log₂M where M is equal to the number of modulation symbols in asignal.
 11. The digital communication system according to claim 10,wherein the signal is modulated as quadrate phase-shifted, 8-ary phaseshift keying, or 16-ary quadrature amplitude modulated.
 12. The digitalcommunication system according to claim 10, wherein the first protographhas n variable nodes and the second protograph has multiple planes andeach labeling bit position from each modulation symbol is mapped to avariable node of one of the planes of the second protograph.
 13. Adigital communication system comprising: a low density parity checkencoder; a bit interleaver; a mapper and modulator; wherein the lowdensity parity check encoder codes binary data with a code having aselected codeword length, and the code is generated by: selecting afirst protograph having a first protograph structure; lifting the firstprotograph to a second protograph by a selected factor; and lifting thesecond protograph to a third protograph via a circulant matrix, whereinthe third protograph provides a protograph code having the selectedcodeword length, and wherein the digital communication system supportssignals having multiple numbers of modulation symbols in a signal,wherein the multiple numbers are represented by M₁, M₂, . . . M_(n) andthe selected factor comprises a common divisor from the solution of theequations shown below for m₁, m₂ . . . m_(n):m₁=log₂M₁, m₂=log₂M₂, . . . , m_(n)=log₂M_(n).
 14. The digitalcommunication system according to claim 10, wherein the first protographis lifted to second protograph by a progressive edge-growth algorithm.15. The digital communication system according to claim 10, wherein thefirst protograph comprises a protograph from AR4JA codes.
 16. Thedigital communication system according to claim 10, wherein the firstprotograph comprises a rate-compatible protograph.
 17. The digitalcommunication system according to claim 10, wherein the first protographcomprises a protograph-based LDPC convolutional code.
 18. The digitalcommunication system according to claim 10, wherein the selected factoris selected to accommodate all modulation schemes within a group ofmodulation schemes.